It All Adds Up: Absolute Value Equations

Italladdsup

Wow! It feels like it has been forever since I last wrote an It All Adds Up post! Luckily, we aren’t too far into the school year, but I’ve got lots of material to cover.

Today, I wanted to expand the concept of Absolute Value. I believe I discussed this in 7th grade, but the principle you always want to remember is: ABSOLUTE VALUE IS ALWAYS POSITIVE! Even if you had |-7| (The absolute value of -7), the answer would be 7. If we had |7|, the answer would still be 7.

However, I want to bring this concept one step further: absolute value equations. What if you had an absolute value inside of an equation? An absolute value with a variable inside of it? Sounds confusing. It is, at first. There is plenty to cover on this topic.

Let’s take this equation:

|x+3| = 7

So how do we solve this? We have to create two separate equations. First, we remove the absolute value sign (for now) and keep the side opposite to the absolute value positive on one side and the same for the second equation but negative. These would be the two equations:

x+3 = 7        x+3 = -7

And so we solve both of them like regular one-step equations and we get the two possible solutions that could equal x. HOWEVER: sometimes we have what is called an extraneous solution. Remember, first, that an absolute value cannot be negative. An extraneous solution is a solution that does not satisfy the original equation.

On another note, we can also have an equation that has no solution. Take this one:

|9x| = -81

We would break both down into the two equations, and we would get the answers: -9 and 9. But how can it be either? An absolute value can not equal a negative number! This would be considered ‘no solution’.

What is we also had a number on the outside of the absolute value as well? Let’s take this equation:

4|4x|=48

Well, we treat the absolute value bars as if they were attached to the number (which they are). So, before we can begin working with the absolute value, we have to perform the normal operation when we would normally be isolating the variable (in this case, we want to isolate the absolute value) For the equation above, we would first divide both sides by 4 and then begin working with |4x|.

What if… we had an absolute value on BOTH sides of the equation? We would first remove the absolute value bars, then make the right side negative for one equation and positive for the second equation. However, for this one, we have to make the ENTIRE right side negative. However, ‘negative’ doesn’t really describe it right: we have to take the inverse of every integer on that side. For example, if we had the equation |5-x| = |-8+x|, our two equations would be these:

5-x = -8 +x       5-x = 8-x

The entire side is not negative: it is just the inverse of the original one (the inverse of -8 is 8). We would solve both equations like the ones with absolute value on one side- but we have to isolate the variable by combing like terms on both sides. Be sure to ALWAYS check for extraneous solutions!

What if there was only a variable inside the absolute value? We solve it the same way! We make sure that the variable is positive, and if there is no number outside of the absolute value, the variable is already isolated!

What if… we dealt with absolute value in inequalities? We would solve them exactly the same way. Just remember: when we divide or multiply both sides by a negative, we must flip the direction of the greater than/lesser than symbol.

Sorry about the great number of ‘what if’s in this post- but there are so many different possible ways to write and solve an absolute value equation- and its best that you know most of them!

Finally, here is this week’s math problem for you to solve. When you have finished, check your answers here.

Solve this absolute value equation:

5|7-x| =  |5x + 5|

Hint: when dividing both sides by a number, divide the numbers inside of the absolute value equation, but keep the bars until you begin solving for x.

Thanks for reading! See you in two weeks for another It All Adds Up post!

Gabe

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